Factoring Exponents?

[HPengineer]

Is it possible to take something like

3^(n+2) - 2^(n+2) and factor out the (n+2)??

[Brian@VeritasPrep]

Hey Engineer,

Good question - and this is a case where I'd recommend that you try to prove the answer to yourself (not because I'm lazy! But because the process of doing that really sharpens your understanding).

Say we factored the ^(n+2) out:

First, it's an exponent and factoring it would detach it from its base, so I don't know what we'd use as its base.

Even at that, we'd then have: [x^(n+2)]*(3-2)

That 3-2 becomes 1, which strips the value off of the bases of 3 and 2. If you do this with potential values of n, you'll see that that doesn't work:

n = 2, then 3^4 - 2^4 = 81 - 16 = 65

Well, 65 should also point something out to us...it doesn't have any exponential factors (it's 13*5...two prime factors different from anything we had to start) so it wouldn't translate to anything multiplied to a power of (n+2) that we could derive from the factored portion.



With exponents, we're essentially good at a few things:

1) Using common bases to equate or combine variables in exponents

2) Multiplying/dividing common bases or common exponents (e.g. 2^2 * 3^2 = (2*3)^2 = 6^2)

3) Finding patterns that occur in exponents (like units digits)

Look for opportunities to do those things and you'll do well...and when you think you're being asked to do something different, prove to yourself whether or not you can (you'll probably have to find a way to do one of the tasks in the list, though).

[HPengineer]

and the light bulb starts to come on thank you sir!

CHeers

M